Nash equilibrium

A Nash equilibrium of a game is a set of (possibly mixed) strategies $\sigma =({\sigma }_1,\mathrm{\dots },{\sigma }_n)$ such that, if each player $i$ believes that that every other player $j$ will play ${\sigma }_j$, then $i$ should play ${\sigma }_i$. That is, when $u_i$ is the utility function for the $i$-th player:

$${\sigma }_i\ne {\sigma }_i^{\prime }\to u_i({\sigma }_i,{\sigma }_{-1})>u_i({\sigma }_i^{\prime },{\sigma }_{-1})$$

$$\forall i\le n\text{ and }\forall {\sigma }_i^{\prime }\in {\mathrm{\Sigma }}_i$$

Translated, this says that if any player plays any strategy other than the one in the Nash equilibrium then that player would do worse than playing the Nash equilibrium.

Title	Nash equilibrium
Canonical name	NashEquilibrium
Date of creation	2013-03-22 12:51:35
Last modified on	2013-03-22 12:51:35
Owner	Henry (455)
Last modified by	Henry (455)
Numerical id	8
Author	Henry (455)
Entry type	Definition
Classification	msc 91A99